New Elements for a Theory of Chaos Topology

Topological analysis in phase space is a fundamental chapter of dynamical systems theory. For deterministic systems, the topological structure of a flow is an invariant that provides information on the mechanisms acting in phase space to shape the flow. A topological analysis involves finding a topological representation of the underlying structure and obtaining an algebraic description allowing for the computation of topological invariants. Developing methods to accomplish these two steps has involved several false starts as well as partially successful roads, which finally lead to what we now call a templex. In this chapter, we introduce the reader to flows in phase space, to the different manners of describing their topological structure, and to the recently developed tools. Examples using ordinary and delay differential equations are discussed.